Discriminants of Hecke algebras

William A. Stein

Abstract:      Using an implementation of the modular symbols algorithm described in Cremona's book Algorithms for modular elliptic curves I computed, for each prime N between 2 and 577, an integer DNwhich is divisible by the discriminant of the Hecke algebra TN associated to weight 2 cusp forms of level N for . For these N we give a table of factored DN.

    The Hecke algebra T=TN is an order in a product E = E1 × E2 × ... × En of totally real number fields. The discriminant of T, denoted disc(T), is the product of the discriminants of the number fields Ei, multiplied by the square of the index of T in its normalization.

Fix a prime number N and let S(N) be the space of weight 2 cusp forms of level N for $\Gamma_0(N)$. For p not equal to N let Tp be the p-th Hecke operator, and let dp be the discriminant of the characteristic polynomial of Tp acting on S(N). Consider the sequence of integers

\begin{displaymath}d_2,\quad \gcd(d_2,d_3), \quad \gcd(d_2,d_3,d_5), \quad \ldots, \quad \gcd(d_2,d_3,\ldots,d_q), \quad \ldots\end{displaymath}

where we omit p if p=N. Since each term divides its predecessor, this sequence must eventually stabilize at some limit $\Delta_N$. Since each term is divisible by the discriminant of $\mathbb T$, this discriminant divides $\Delta_N$.

      I have written a program which computes the above sequence until it repeats some value DN for 15 terms. The result of that computation is given in table 1, which can be found at the end of this document. It is interesting to note that N=389 is the only case in our tables for which N|DN. I have checked up to N=14537 and found no other cases in which this occurs. Whether or not this ever occurs is of interest to Ribet as this hypothesis plays a role in his paper, ``Torsion points on J0(N) and galois representations.''

      Another problem is to determine, for each N in table 1, whether the primes dividing DN are exactly the same as the primes dividing disc(T). I have checked that this is the case for N < 73. If the ring $\mathbb T\otimes\mathbb F_p$ is not reduced then p|disc(T). This ring can't be reduced if Tq is not diagonalizable (modulo p) for some prime q not equal to N. However, this sufficient condition is not always necessary, as the case N=37 illustrates. Here 2 ramifies in the Hecke algebra even though the Hecke operators Tq with $q\not=2$ act semisimply modulo 2.


Table of factored DN for N<577

  N DN   (upper bound on discriminant of Hecke algebra)
11 1
13 0
17 1
19 1
23 5
29 23
31 5
37 22
41 22×37
43 25
47 19×103
53 24×37
59 27×31×557
61 24×37
67 24×54
71 34×2572
73 24×32×5×13
79 24×83×983
83 28×197×11497
89 26×53×6689
97 26×72×2777
101 28×17568767
103 28×5×17×411721
107 212×5×7×1667×19079
109 210×72×7537
113 210×34×72×112×107
127 212×34×7×86235899
131 219×5×46141×75619573
137 210×52×29×401×895241
139 214×32×72×997×2151701
149 212×72×234893×1252037
151 218×72×11×672×257×439867
157 213×61×397×48795779
163 215×32×65657×82536739
167 216×5×8269×5103536431379173
173 214×52×7×29×5608385124289
179 222×34×72×313×137707×536747147
181 216×52×7×61×397×595051637
191 28×33×5×382146223×319500117632677
193 214×5×112×17×103×401×4153×680059
197 218×52×61×397×35217676193989
199 216×3×53×29×31×712×347×947×37316093
211 220×3×5×74×412×43×229×52184516509
223 236×72×19×103×3995922697473293141
227 237×32×53×74×132×29×312×13591×57139×273349
229 232×107×17467×39555937×53625889
233 222×37×53×139×653×4127×24989×8388019
239 212×72×2833×51817×97423×1174779433×8920940047
241 223×97×1489×20857×651474368435017
251 228×52×29×373×8768135668531×2006012696666681
257 265×29×479×71711×409177×654233×32354821
263 220×11×61×397×15631853×34867513×97092067×252746489
269 222×32×43×151×27767×65657×5550873754172978311
271 224×32×1367×6091×592661×1132673×14171513×172450541
277 222×52×19×29×37×1372×92767×1530091×25531570859
281 222×3×5×181×857×8388019×2647382149×1778899342669
283 246×349×1297×413713×73199099×5832488839
293 226×32×29×233×23512×69763×42711913589792108923
307 250×36×55×112×133×1072×457×3697×21577×974513×568380457
311 216×52×29×3013091897×2106873009119126062143259000543887593
313 224×5×412×8619587×9614923×130838023×2164322751511
317 226×7×367×3217×660603043×14989400036918065702697531
331 238×32×532×229×1399×21911×205493×6363601×584461573862449
337 228×113×593×2791×2963615537×747945736667×4122851467451
347 261×5×72×192×331×349×479×617×1797330450291217×918291275915301361
349 228×13×103×1118857×72318613×6771977049413×1313981654817031
353 234×32×5×1272×229×114641×551801×12611821×7779730837×24314514437
359 236×36×2777×16512254293×64542630435970307×2171776478013633068927
367 244×7×81421×251387×418175501×15354151381×13144405392643360366681
373 232×7×113×23×199×673×2143×1542194372227×72819251148518000363297
379 234×59×317×421×278329×5698591×2117788336277×2851210737989187265253
383 232×5×112×13×72893×3151861×16141144314299× 178236551484825400362837637090811
389 263×34×56×312×37×389×3881×215517113148241×477439237737571441
397 257×232×312×97×317×7612×302609750073209×83566618884497478937
401 296×52×19×163×2932×811×1218675071×71742740351×388881803749×34393898968391
409 232×33×17×1667×1741×2341×537071×14884451×18631199×1334964067081334453235547
419 255×17×43×113×151×167×971×493657×20375986548898473293×53097073649092855361102575237
421 234×3×31×557×4729×825403×857459×144211946777593109 ×2328579379136648917067
431 291×34×56×11×192×29×31×43×197×257×69472×37619×29252013842927×806505757406715084824003
433 268×37×72×372×101×379×1439×3613×18719×2792477×77087971×5830108671536745647
439 266×32×5×312×173×84179×85667×16794662617×513841517138871835091506167235408934202857
443 288×32×72×312×499×6899×48508479390300197×2817219327571188909266947704801865987
449 240×3×72×101×44933757980789×188247485945671 ×653016225615601×1431966252229376199841
457 236×5×312×653×3169×38983093×52621913×33122975406370693×5653726203394180386934181
461 273×5×72×193×972×80750473×3104029729×607263139073×3729490905341009668647473177
463 262×113×311×9929×568201×132502583×1474412920219×2770309905285622039024420194209857723
467 271×172×1212648089519×32432206859088781×6296651104824906148358708614333895055221783
479 232×13×17×1861×4021×28745083×41556253×1202203127423×201529385024397103×7037463122648759781611869895003
487 272×316×54×132×172×194×59×1032×109×257×623519211698413571686763×15408475904697077364866629
491 2104×56×192×43×131×479×887×5650859×54796097920639362740205317747356273097682333252495603721
499 269×311×5×712×167×495613×25224990196319×573452584782809×277143583167463430555979797274731
503 278×32×54×112×193×257×821×20032×13597×45587×384479819×8659024393×20115672029938390602701696607766073563
509 271×33×13×157×971×1277×4567×3691783×42330311×1157039662523351992921397×6331071860925306189417509
521 242×23×53×67×929×13877×531096383×19526270957×1089951135204631559833×14340527343875384245648725589439
523 291×3×5×413×59×1492×1201×279121937×8371971617×9059602909494267071628228952878552757512056969593
541 246×32×5×13×277×307×591581×1940573213×221136462575339×1453183329662653×18044474614550745414465332996771
547 2105×73×73×1032×5501×11783×16097×43781×1152631×146768003×9959758037×91268351929×102277460687×106666343972273
557 246×74×132×4787×252163×16849164271275021852893×53296770296923102812608983×2381022539751738307256162767
563 2139×52×134×372×612×37591×52667×155083×301703×938251×46706589087295134421×299128314984453465128592656821021
569 246×73×449531828286229614392569×189316003×257022598600391962761793946239×2294643649486046267496627432517
571 2166×312×58×74×132×17×373×412×792×1272×181×211×293×709×15792×16672×12030433×807024744595934649052018211
577 2131×312×54×133×592×612×257×163753×41340850017998228328234516909328723846661×85934741209775683850815667